ISBN: 3-540-65941-2
TITLE: Applied Finite Group Actions
AUTHOR: Kerber, Adalbert
TOC:

Preface to the Second Edition vii
Preface to the First Edition ix
List of Symbols xv
0. Labeled Structures 1
0.1 Species of Structures 2
0.2 Sum and Product of Species 10
0.3 Partitional Composition 13
0.4 Derivation, Pointing, Functorial Composition 16
0.5 The Ring of Isomorphism Classes of Species 19
1. Unlabeled Structures 21
1.1 Group Actions 21
1.2 Orbits, Cosets and Double Cosets 29
1.3 Symmetry Classes of Mappings 36
1.4 Invariant Relations 45
1.5 Hidden Symmetries 50
2. Enumeration of Unlabeled Structures 53
2.1 The Number of Orbits 53
2.2 Enumeration of Symmetry Classes 59
2.3 Application to Incidence Structures 67
2.4 Special Symmetry Classes 73
3. Enumeration by Weight 85
3.1 Weight Functions. 85
3.2 Cycle Indicator Polynomials 91
3.3 Sums of Cycle Indicators, Recursive Methods 100
3.4 Applications to Chemistry 103
3.5 A Generalization 109
3.6 The Decomposition Theorem 115
4. Enumeration by Stabilizer Class 121
4.1 Counting by Stabilizer Class 121
4.2 Asymmetric Orbits, Lyndon Words, the Cyclotomic Identity 126
4.3 Tables of Marks and Burnside Matrices 131
4.4 Weighted Enumeration by Stabilizer Class 137
5. Poset and Semigroup Actions 141
5.1 Actions on Posets, Semigroups, Lattices 141
5.2 Examples 150
5.3 Application to Combinatorial Designs 157
5.4 The Burnside Ring 162
6. Representations 169
6.1 Representations of Symmetric Groups 169
6.2 Tableaux and Matrices 181
6.3 The Determinantal Form 187
6.4 Standard Bideterminants 194
7. Further Applications 213
7.1 Schur Polynomials 213
7.2 Symmetric Polynomials 219
7.3 The Diagram Lattice 223
7.4 Unimodality 228
7.5 The LittlewoodRichardson Rule 236
7.6 The MurnaghanNakayama Rule 247
7.7 Symmetrization and Permutrization 255
7.8 Plethysm of Representations 260
7.9 Actions on Chains 267
8. Permutations 275
8.1 Multiply Transitive Groups 275
8.2 Root Number Functions 284
8.3 Equations in Groups 293
8.4 UpDown Sequences 297
8.5 Foulkes Characters 303
8.6 Schubert Polynomials 307
9. Construction and Generation 317
9.1 Orbit Evaluation 318
9.2 Transversals of Symmetry Classes 321
9.3 Orbits of Centralizers 326
9.4 The Homomorphism Principle 329
9.5 Orderly Generation 334
9.6 Generating Orbit Representatives 337
9.7 Symmetry Adapted Bases 341
9.8 Applications of Symmetry Adapted Bases 346
10. Tables 353
10.1 Tables of Marks and Burnside Matrices 353
10.1.1 Cyclic Groups 354
10.1.2 Dihedral Groups 358
10.1.3 Alternating Groups 363
10.1.4 Symmetric Groups 366
10.2 Characters of Symmetric Groups 369
10.2.1 Irreducible Characters and Young Characters 369
10.2.2 Foulkes Tables 378
10.2.3 Character Polynomials 380
10.3 Schubert Polynomials 392
11. Appendix 397
11.1 Groups 397
11.2 Finite Symmetric Groups 399
11.3 Rothe Diagram and Lehmer Code 406
11.4 Linear Representations 412
11.5 Ordinary Characters of Finite Groups 418
11.6 The Mbius Inversion 424
12. Comments and References 429
12.1 Historical Remarks, Books and Review Articles 429
12.2 Further Comments 432
12.3 Suggestions for Further Reading 433
References 437
Index 445
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